Methods for determining the orbital angular momentum of a laser beam
Kotlyar V.V., Kovalev A.A., Porfirev A.P.


IPSI RAS - Branch of the FSRC “Crystallography and Photonics” RAS, 443001, Samara, Russia, Molodogvardeyskaya 151
Samara National Research University, 443086, Russia, Samara, Moskovskoye Shosse 34


We propose and study numerically and experimentally two methods for determining the orbital angular momentum (OAM) of paraxial laser beams. One method is based on registering the intensity in the Fresnel zone, numerically averaging this intensity over the polar angle at discrete radii, and solving a system of linear equations to find squared modules of the coefficients of the light field expansion in terms of basis functions. The other method is based on registering two intensity distributions in the Fourier plane of two cylindrical lenses rotated 90 degrees relative to each other, and calculating the first-order moments of the measured intensities. The experimental error of the OAM determination is about 1% for small fractional OAM (up to 4), and about 8% for large fractional OAM (up to 30).

paraxial laser beam, fractional orbital angular momentum, fractional topological charge, optical vortex, cylindrical lens, superposition of spatial modes.

Kotlyar VV, Kovalev AA, Porfirev AP. Methods for determining the orbital angular momentum of a laser beam. Computer Optics 2019; 43(1): 42-53. DOI: 10.18287/2412-6179-2019-43-1-42-53.


  1. Wang J. Advances in communications using optical vortices. Phot Res 2016; 4(5): B14-B28. DOI: 10.1364/PRJ.4.000B14.
  2. Wang Z, Yan Y, Arbabi A, Xie G, Liu C, Zhao Z, Ren Y, Li L, Ahmed N, Willner AJ, Arbabi E, Faraon A, Bock R, Ashrafi S, Tur M, Willner AE. Orbital angular momentum beams generated by passive dielectric phase masks and their performance in a communication link. Opt Lett 2017; 42(14): 2746-2749. DOI: 10.1364/OL.42.002746.
  3. Wang X, Nie Z, Liang Y, Wang J, Li T, Jia B. Recent advances on optical vortex generation. Nanophotonics 2018; 7(9): 1533-1556. DOI: 10.1515/nanoph-2018-0072.
  4. Gbur GJ. Singular optics. Boca Raton: CRC Press; 2016. ISBN: 978-1-4665-8077-0.
  5. Kotlyar VV, Kovalev AA, Porfirev AP. Vortex laser beams. Boca Raton: CRC Press; 2018. ISBN: 978-1-1385-4211-2.
  6. Khonina SN, Kotlyar VV, Skidanov RV, Soifer VA, Laakkonen P, Turunen J, Wang Y. Experimental selection of spatial Gauss-Laguerre modes. Optical Memory & Neural Networks 2000; 9(1): 73-82.
  7. Khonina SN, Kotlyar VV, Soifer VA, Pääkkönen P, Simonen J, Turunen J. An analysis of the angular momentum of a light field in terms of angular harmonics. J Mod Opt 2001; 48(10): 1543-1557.– DOI: 10.1080/09500340108231783.
  8. Ruffato G, Massari M, Parisi G, Romanato F. Test of mode-division multiplexing and demultiplexing in free-space with diffractive transformation optics. Opt Express 2017; 25(7): 7859-7868. DOI: 10.1364/OE.25.007859.
  9. Li Y, Li X, Chen L, Pu M, Jin J, Hong M, Luo X. Orbital angular momentum multiplexing and demultiplexing by a single metasurface. Adv Opt Mater 2016; 5(2): 1600502. DOI: 10.1002/adom.201600502.
  10. Bekshaev AYa, Soskin MS, Vasnetsov MV. Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams. J Opt Soc Am A 2003; 20(8): 1635-1643. DOI: 10.1364/JOSAA.20.001635.
  11. Denisenko V, Shvedov V, Desyatnikov AS, Neshesv DN, Krolikowski W, Volyar A, Soskin MS, Kivshar YS. Determination of topological charges of polychromatic optical vortices. Opt Express 2009; 17(26): 23374-23379. DOI: 10.1364/OE.17.023374.
  12. Kotlyar VV, Kovalev AA, Porfirev AP. Astigmatic transforms of an optical vortex for measurement of its topological charge. Appl Opt 2017; 56(14): 4095-4110. DOI: 10.1364/AO.56.004095.
  13. Alperin SN, Niederriter RD, Gopinath JT, Siemens ME. Quantitative measurement of the orbital angular momentum of light with a single, stationary lens. Opt Lett 2016; 41(21): 5019-5022. DOI: 10.1364/OL.41.005019.
  14. Alperin SN, Siemens ME. Angular momentum of topologically structured darkness. Phys Rev Lett 2017; 119(20): 203902. DOI: 10.1103/PhysRevLett.119.203902.
  15. Liu Z, Gao S, Xiao W, Yang J, Huang X, Feng Y, Li J, Liu W, Li Z. Measuring high-order optical orbital angular momentum with a hyperbolic gradually changing period pure-phase grating. Opt Lett 2018; 43(13): 3076-3079. DOI: 10.1364/OL.43.003076.
  16. Maji S, Brundavanam MM. Controlled noncannonical vortices from higher-order fractional screw dislocations. Opt Lett 2017; 42(12): 2322-2325. DOI: 10.1364/OL.42.002322.
  17. D'Errico A, D'Amelio R, Piccirillo B, Cardano F, Marrucci L. Measuring the complex orbital angular momentum spectrum and spatial mode decomposition of spectrum light beams. Optica 2017; 4(11): 1350-1357. DOI: 10.1364/OPTICA.4.001350.
  18. Kotlyar VV, Kovalev AA, Porfirev AP. Astigmatic laser beams with a large orbital angular momentum. Opt Express 2018; 26(1): 141-156. DOI: 10.1364/OE.26.000141.
  19. Melo LA, Jesus-Silva AJ, Chavez-Cedra S, Ribeiro PHS, Soares WC. Direct measurement of the topoloigical charge in elliptical beams using diffraction by a triangular aperture. Sci Rep 2018; 8: 6370. DOI: 10.1038/s41598-018-24928-5.
  20. Gao H, Han Y, Li Y, Zhu D, Sun M, Yu S. Topological charge measurement of concentric OAM states using the phase-shift method. J Opt Soc Am A 2018; 35(1): A40-A44. DOI: 10.1364/JOSAA.35.000A40.
  21. Volyar AV, Bretsko MV, Akimova YaE, Egorov YuA. Beyond the light intensity or intensity moments and measurements of the vortex spectrum in complex light beams [In Russian]. Computer Optics 2018; 42(5): 736-743. DOI: 10.18287/2412-6179-2017-42-5-736-743.
  22. Siegman AE. Lasers. Mill Valley, CA: University Science Books; 1986. ISBN: 978-0-935702-11-8.
  23. Xie Z, Gao S, Lei T, Feng S, Zhang Y, Li F, Zhang J, Li Z, Yuan X. Integrated (de)multiplexer for orbital angular momentum fiber communication. Phot Res 2018; 6(7): 743-749. DOI: 10.1364/PRJ.6.000743.
  24. Gori F, Guattari G, Padovani C. Bessel-Gauss beams. Opt Commun 1987; 64(6): 491-495. DOI: 10.1016/0030-4018(87)90276-8.
  25. Berry MV. Optical vortices evolving from helicoidal integer and fractional phase steps. J Opt A: Pure Appl Opt 2004; 6(2): 259-268. DOI: 10.1088/1464-4258/6/2/018.
  26. Gotte JB, Franke-Arnold S, Zambrini R, Barnett SM. Quantum formulation of fractional orbital angular momentum. J Mod Opt 2007; 54(12): 1723-1738. DOI: 10.1080/09500340601156827.
  27. Kotlyar VV, Kovalev AA, Soifer VA. Asymmetric Bessel modes. Opt Lett 2014; 39(8): 2395-2398. DOI: 10.1364/OL.39.002395.
  28. Prudnikov AP, Brychkov YA, Marichev OI. Integrals and series. Volume 2: Special functions. New York: Gordon and Breach; 1986. ISBN: 2-88124-097-6.

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